Data Analysis Using Regression and Multilevel/Hierarchical Models

Logistic regression is the standard way to model binary outcomes (that is, data yi that take on the values 0 or 1). Section 5.1 introduces logistic regression in a simple example with one predictor, then for most of the rest of the chapter we work through an extended example with multiple predictors and interactions.

Logistic regression with a single predictor

Example: modeling political preference given income

Conservative parties generally receive more support among voters with higher incomes. We illustrate classical logistic regression with a simple analysis of this pattern from the National Election Study in 1992. For each respondent i in this poll, we label yi = 1 if he or she preferred George Bush (the Republican candidate for president) or 0 if he or she preferred Bill Clinton (the Democratic candidate), for now excluding respondents who preferred Ross Perot or other candidates, or had no opinion. We predict preferences given the respondent's income level, which is characterized on a five-point scale.

The data are shown as (jittered) dots in Figure 5.1, along with the fitted logistic regression line, a curve that is constrained to lie between 0 and 1. We interpret the line as the probability that y = 1 given x—in mathematical notation, Pr(y = 1|x).